System and Method for Tracking an Expanded State of a Moving Object Using a Compound Measurement Model

ABSTRACT

A tracking system for tracking an expanded state of an object is provided. The tracking system comprises at least one processor and a memory having instructions stored thereon that, when executed by the at least one processor, cause the tracking system to execute a probabilistic filter that iteratively tracks a belief on the expanded state of the object, wherein the belief is predicted using a motion model of the object and is further updated using a compound measurement model of the object. The compound measurement model includes multiple probabilistic distributions constrained to lie on a contour of the object with a predetermined relative geometrical mapping to the center of the object. Further, the tracking system tracks the expanded state of the object based on the updated belief on the expanded state.

TECHNICAL FIELD

The present disclosure relates generally to automotive object tracking,and more specifically to a system and a method for tracking an expandedstate of an object using measurements of the object.

BACKGROUND

Control systems employed by vehicles, such as autonomous vehicles andsemi-autonomous vehicles, predict safe motion or path for the vehiclesto avoid collision with obstacles, such as other vehicles orpedestrians. In some scenarios, a vehicle is also configured for sensingits surroundings, such as road edges, pedestrians, and other vehicles,with the help of one or more sensors of the vehicle. Some of thesesensors include ultrasonic sensors, cameras, and LIDAR sensors, whichare used in existing advanced driver assistance systems (ADAS).

The control system of the vehicle tracks an object state of the othervehicles (where the object state includes kinematic states) based onautomotive radar measurements, to control the vehicle. Extended objecttracking (EOT) with multiple measurements per scan has shown improvedobject tracking than the traditional point object tracking whichincludes only one measurement per scan, by augmenting the object statefrom kinematic-only state to both kinematic and extended states. Theextended state provides dimension and orientation of the objects undertracking. To achieve this, spatial distribution (i.e. how automotiveradar measurements are spatially distributed around the object) needs tobe captured along with sensor noise. Current methods include a frameworkof a fixed set of points on a rigid body that requires a non-scalabledata association between the fixed set of points and automotive radardetections even for a single object tracking. Spatial models, such asthe contour model and surface model, bypass the cumbersome dataassociation step.

For automotive radar measurements, the contour model reflects themeasurement distribution along the contour of an object (e.g., the rigidbody), and the surface model assumes the radar measurements aregenerated from the inner surface of a two-dimensional shape. Examples ofthe contour model include a simple rectangular shape and a more generalstar-convex shape modeled by either a random hypersurface model or aGaussian process model. Some surface models such as Gaussian-basedellipse and hierarchical Gaussian-based ellipse model arecomputationally much simpler than the contour model that requires muchmore degrees of freedom to describe a more complex shape. However, themeasurements of the object are subject to noise, and reflections arereceived only from the surface of the object. Therefore, theaforementioned models don't capture real-world automotive radarmeasurements.

Accordingly, there is a need for a system and a method for tracking boththe kinematic and extended states of the object by capturing thereal-world automotive radar measurements.

SUMMARY

It is an object of some embodiments to provide a system and a method fortracking an expanded state of an object. The expanded state of theobject includes a kinematic state indicative of one or a combination ofa position and a velocity of a center of the object, and an extendedstate indicative of one or a combination of a dimension and anorientation of the object. The center of the object is one or acombination of an arbitrarily selected point, a geometrical center ofthe object, a center of gravity of the object, a center of a rear axisof wheels of a vehicle, and the like. A sensor, for example, automotiveradar, is used to track objects (such as a vehicle). In an embodiment,the automotive radar may provide direct measurements of radialvelocities, long operating ranges, small sizes at millimeter orsub-terahertz frequency bands, and high spatial resolutions.

In point object tracking, a single measurement per scan is received fromthe vehicle. The point object tracking provides only the kinematic state(position) of the vehicle. Further, a probabilistic filter with ameasurement model having distribution of kinematic states is utilized totrack the vehicle. In extended object tracking (EOT), multiplemeasurements per scan are received. The multiple measurements arespatially structured around the vehicle. The extended object trackingprovides both the kinematic and the extended state of the vehicle. Theprobabilistic filter with a measurement model having distribution ofextended states is utilized to track the vehicle.

However, real-world automotive radar measurement distributions show thatmultiple reflections from the vehicle are complex. Due to thiscomplexity, designing a proper measurement model becomes complex.Therefore, regular measurement models are applicable only for thekinematic state and not for the expanded state.

To that end, in some embodiments, spatial models such as a contour modeland a surface model are used to capture the real-world automotive radarmeasurements. However, the aforesaid spatial models are inaccurate. Someembodiments are based on the recognition that real-world automotiveradar measurements are distributed around edges or the surface of theobjects (the vehicle) with a certain volume, which gives rise to asurface volume model. To that end, some embodiments are based on theobjective of formulating a surface volume model that resembles andcaptures the real-world automotive radar measurements. The surfacevolume model balances between the contour model and the surface modelwith more realistic features while keeping the EOT accurate.

In particular, in an embodiment, based on principles of the contourmodel and the surface model, a compound measurement model (which is atype of surface volume model) is determined. The compound measurementmodel includes multiple probabilistic distributions constrained to lieon a contour of the object with a predetermined relative geometricalmapping to the center of the object. The multiple probabilisticdistributions are used to cover a measurement spread along the contourof the object.

The compound measurement model is compound in multiple ways. Forexample, the compound measurement model has a compound structure, i.e.,the multiple probabilistic distributions. Also, the compound measurementmodel has a compound composition, i.e., functions of the multipleprobabilistic distributions, a function of the contour, and theirrelationship. Further, the compound measurement model has a compoundnature, i.e., the multiple probabilistic distributions are based on themeasurements and thus represent data-driven approaches of modelgeneration, whereas the contour is based on modeling a shape of theobject, e.g., a shape of a vehicle, using principles of physics-basedmodeling.

Additionally, the compound measurement model takes advantage ofdifferent principles of modeling the expanded state, i.e., the compoundmeasurement model joins the principles of the contour model and thesurface model. As a result, the compound measurement model betterrepresents the physical nature of tracking the object while simplifyingmeasurement assignment. In addition, the multiple probabilisticdistributions of the compound measurement model are more flexible over asingle distribution of the surface model, and can better describe thecontour of the object, and are more flexible to explain the measurementscoming from different angles or views of the object.

Some embodiments are based on understanding that, in theory, themultiple probabilistic distributions can lie on the contour, assumingthat a shape of the contour has no restrictions. However, in practice,such assumptions are incorrect for tracking the expanded state. Incontrast, the contour of the object is predetermined and the multipleprobabilistic distributions are fit to the contour rather than thecontour is fit to the multiple probabilistic distributions.

The compound measurement model is learned offline, i.e., in advance. Thecompound measurement model may be learned in a unit coordinate system ora global coordinate system. Some embodiments are based on recognitionthat it is beneficial to learn the compound measurement model in theunit coordinate system, because it simplifies the calculation and makesthe compound measurement model agnostic to the dimensions of the object.Each of the multiple probabilistic distributions (represented asellipses) can be assigned with measurements in a probabilistic manner.The measurements associated with the ellipse may be referred to asellipse-assigned measurements.

According to some embodiments, the offline learned compound measurementmodel is used for online tracking of the expanded state of the object,i.e., real time tracking of the expanded state of the object. Someembodiments are based on the realization that the probabilistic natureof the compound measurement model can be beneficially aligned withprobabilistic multi-hypothesis tracking (PMHT) methods. For example,such an alignment allows implementing the probabilistic filter using atleast a variation of a Kalman filter. For example, one embodiment usesan unscented Kalman filter-probabilistic multi-hypothesis tracking(UKF-PMHT) method. The unscented Kalman filter (UKF) is used fortransforming the compound measurement model from the unit coordinatesystem into the global coordinate system. The probabilisticmulti-hypothesis tracking (PMHT) method is then applied to assignmeasurements at a current time step to different probabilisticdistributions in a probabilistic fashion, and update the expanded stateof the object.

According to an embodiment, given an expanded state of the object and acovariance matrix corresponding to a previous time step, and a motionmodel of the object, an expanded state of the object for a current timestep and a covariance matrix corresponding to the expanded state can bepredicted. In an embodiment, the motion model may be a coordinated turn(CT) motion model with polar velocity. In some other embodiments, forthe kinematic states, the coordinated turn (CT) motion model with polarvelocity is used and, for the extended state, i.e., length and width, aconstant model is used with a process noise with a small covariance asthe length and width are unlikely changed over time. The predictedexpanded state of the object may be referred to as a predicted belief ofthe expanded state because this prediction is probabilistic. Someembodiments are based on recognition that the predicted expanded stateof the object may be inaccurate. To correct the predicted expanded stateof the object, the compound measurement model in a unit coordinatesystem is used. However, the predicted expanded state is in a globalcoordinate system. Therefore, to align the compound measurement model inthe unit coordinate system with the predicted expanded state, thecompound measurement model needs to be transformed from the unitcoordinate system to the global coordinate system. In particular, theellipse-assigned measurements in the unit coordinate system need to betransformed into the global coordinate system.

Some embodiments are based on realization that such a transformation canbe achieved using an unscented transform function (or UKF). To that end,in an embodiment, sigma points are generated for an ellipse (i.e., for aprobabilistic distribution of the compound measurement model). The“ellipse” and “probabilistic distribution” may be used interchangeablyand would mean the same. Further, the sigma points are propagated intothe unscented transform function and, consequently, predictedmeasurements in the global coordinate system corresponding to theellipse-assigned measurements of the ellipse in the unit coordinatesystem is determined. Additionally, a covariance matrix corresponding tothe predicted measurements is determined based on the predictedmeasurements. Likewise, the measurements in the global coordinate systemcorresponding to the ellipse-assigned measurements associated with therest of the ellipses are determined. To that end, a predicted expandedstate model, where the compound measurement model is aligned accordingto the predicted expanded state, is obtained.

Further, the measurements at the current time step are received. Someembodiments are based on the realization that the multiple probabilisticdistributions (ellipses) can be treated independently of each other.Such an independent treatment allows considering different view-anglesfor probing the expanded state of the object. To consider such anindependent treatment, some embodiments treat different probabilisticdistributions of the multiple probabilistic distributions as belongingto different objects. Additionally, some embodiments are based on therealization that a soft probabilistic assignment, i.e., probabilisticassignment of the measurements to different probabilistic distributions,is more advantageous than a hard deterministic assignment. To that end,the measurements are assigned to each probabilistic distribution with acorresponding association probability. The measurements with thecorresponding association probability associated with each probabilisticdistribution are referred to as the ‘synthetic measurements’.

Further, in some embodiments, a synthetic centroid and a syntheticcovariance matrix are determined for each probabilistic distributionbased on the corresponding synthetic measurements. Further, using thesynthetic measurements associated with each probabilistic distribution,the predicted belief on the expanded state is updated. For instance, thepredicted belief on the expanded state may be updated using theprobabilistic filter, such as the Kalman filter, with the syntheticmeasurements associated with each probabilistic distribution to producean updated expanded state of the object.

Accordingly, one embodiment discloses a tracking system for tracking anexpanded state of an object including a kinematic state indicative of acombination of a position and a velocity of a center of the object andan extended state indicative of a combination of a dimension and anorientation of the object. The tracking system comprises: at least oneprocessor; and memory having instructions stored thereon that, whenexecuted by the at least one processor, cause the tracking system toreceive measurements associated with at least one sensor, wherein atleast one sensor is configured to probe a scene including the objectwith one or multiple signal transmissions to produce one or multiplemeasurements of the object per the transmission; execute a probabilisticfilter iteratively tracking a belief on the expanded state of theobject, wherein the belief is predicted using a motion model of theobject and is updated using a compound measurement model of the object,wherein the compound measurement model includes multiple probabilisticdistributions constrained to lie on a contour of the object with apredetermined relative geometrical mapping to the center of the object,wherein in each iteration of the iterative tracking, the belief on theexpanded state is updated based on a difference between a predictedbelief and an updated belief, wherein the updated belief is estimatedbased on probabilities of the measurements fitting each of the multipleprobabilistic distributions, and mapped to the expanded state of theobject based on the corresponding geometrical mapping; and track theexpanded state of the object based on the updated belief on the expandedstate.

Accordingly, another embodiment discloses a tracking method for trackingan expanded state of an object including a kinematic state indicative ofone or a combination of a position and a velocity of a center of theobject and an extended state indicative of one or a combination of adimension and an orientation of the object. The tracking methodcomprises receiving measurements associated with at least one sensor,wherein at least one sensor is configured to probe a scene including theobject with one or multiple signal transmissions to produce one ormultiple measurements of the object per the transmission; executing aprobabilistic filter iteratively tracking a belief on the expanded stateof the object, wherein the belief is predicted using a motion model ofthe object and updated using a compound measurement model of the object,wherein the compound measurement model includes multiple probabilisticdistributions constrained to lie on a contour of the object with apredetermined relative geometrical mapping to the center of the object,wherein in each iteration of the iterative tracking, the belief on theexpanded state is updated based on a difference between a predictedbelief and an updated belief, wherein the updated belief is estimatedbased on probabilities of the measurements fitting each of the multipleprobabilistic distributions, and mapped to the expanded state of theobject based on the corresponding geometrical mapping; and tracking theexpanded state of the object based on the updated belief on the expandedstate.

A non-transitory computer readable storage medium embodied thereon aprogram executable by a processor for performing a method for trackingan expanded state of an object, wherein the expanded state includes akinematic state indicative of one or a combination of a position and avelocity of a center of the object and an extended state indicative ofone or a combination of a dimension and an orientation of the object.The method comprises receiving measurements associated with at least onesensor, wherein at least one sensor is configured to probe a sceneincluding the object with one or multiple signal transmissions toproduce one or multiple measurements of the object per the transmission;executing a probabilistic filter iteratively tracking a belief on theexpanded state of the object, wherein the belief is predicted using amotion model of the object and is updated using a compound measurementmodel of the object, wherein the compound measurement model includesmultiple probabilistic distributions constrained to lie on a contour ofthe object with a predetermined relative geometrical mapping to thecenter of the object, wherein in each iteration of the iterativetracking, the belief on the expanded state is updated based on adifference between a predicted belief and an updated belief, wherein theupdated belief is estimated based on probabilities of the measurementsfitting each of the multiple probabilistic distributions, and mapped tothe expanded state of the object based on the corresponding geometricalmapping; and tracking the expanded state of the object based on theupdated belief on the expanded state.

BRIEF DESCRIPTION OF THE DRAWINGS

The presently disclosed embodiments will be further explained withreference to the attached drawings. The drawings shown are notnecessarily to scale, with emphasis instead generally being placed uponillustrating the principles of the presently disclosed embodiments.

FIGS. 1A, 1B and 1C collectively show a schematic overview of principlesfor tracking an expanded state of an object, according to someembodiments.

FIG. 2 shows a block diagram of a tracking system for tracking theexpanded state of the object, according to some embodiments.

FIGS. 3A, 3B, and 3C collectively show various schematics for tracking abelief on the expanded state of the object using a compound measurementmodel, according to some embodiments.

FIG. 4 shows a flow chart of a method for learning parameters of thecompound measurement model, according to some embodiments.

FIGS. 5A and 5B show schematics of transformation of training datacollected from different motions of different objects into a common unitcoordinate system, according to some embodiments.

FIG. 6 shows a block diagram of an expectation-maximization (EM) methodfor learning the parameters of the compound measurement model, accordingto some embodiments.

FIG. 7A shows a flowchart of an unscented Kalman filter-probabilisticmulti-hypothesis tracking (UKF-PMHT) algorithm, according to someembodiments.

FIG. 7B shows a block diagram of steps performed to compute predictedmeasurements and a covariance matrix, according to some embodiments.

FIG. 7C shows a block diagram of steps performed to compute syntheticmeasurements and a synthetic covariance matrix, according to someembodiments.

FIG. 7D shows a block diagram of steps performed to update the expandedstate and a covariance matrix, according to some embodiments.

FIG. 8A shows a schematic of a vehicle including a controller incommunication with the system employing principles of some embodiments.

FIG. 8B shows a schematic of interaction between the controller of thesystem of FIG. 8A and controllers of the vehicle, according to someembodiments.

FIG. 8C shows a schematic of an autonomous or semi-autonomous controlledvehicle for which control inputs are generated by using someembodiments.

DETAILED DESCRIPTION

In the following description, for purposes of explanation, numerousspecific details are set forth in order to provide a thoroughunderstanding of the present disclosure. It will be apparent, however,to one skilled in the art that the present disclosure may be practicedwithout these specific details. In other instances, apparatuses andmethods are shown in block diagram form only in order to avoid obscuringthe present disclosure.

As used in this specification and claims, the terms “for example,” “forinstance,” and “such as,” and the verbs “comprising,” “having,”“including,” and their other verb forms, when used in conjunction with alisting of one or more components or other items, are each to beconstrued as open ended, meaning that that the listing is not to beconsidered as excluding other, additional components or items. The term“based on” means at least partially based on. Further, it is to beunderstood that the phraseology and terminology employed herein are forthe purpose of the description and should not be regarded as limiting.Any heading utilized within this description is for convenience only andhas no legal or limiting effect.

FIGS. 1A, 1B and 1C show a schematic overview of some principles used bysome embodiments for tracking an expanded state of an object. Theexpanded state of the object includes a kinematic state indicative ofone or a combination of a position and a velocity of a center of theobject, and an extended state indicative of one or a combination of adimension and an orientation of the object. The center of the object isone or a combination of an arbitrarily selected point, a geometricalcenter of the object, a center of gravity of the object, a center of arear axis of wheels of a vehicle, and the like. A sensor 104 (forexample, automotive radar) is used to track objects (such as vehicle106). In point object tracking 100, a single measurement 108 per scan isreceived from the vehicle 106. The point object tracking 100 providesonly the kinematic state (position) of the vehicle 106. Further, aprobabilistic filter with a measurement model having distribution ofkinematic states is utilized to track the vehicle 106. In extendedobject tracking (EOT) 102, multiple measurements 110 per scan arereceived. The multiple measurements 110 are spatially structured aroundthe vehicle 106. The EOT 102 provides both the kinematic and theextended state of the vehicle 106. The probabilistic filter with ameasurement model having distribution of extended states is utilized totrack the vehicle 106.

However, real-world automotive radar measurement 112 distributions, asillustrated in FIG. 1B, show that multiple reflections from the vehicle106 are complex. Due to this complexity, designing of the measurementmodel becomes complex. Therefore, regular measurement models areapplicable only for the kinematic state and not for the expanded state.

To that end, in some embodiments, spatial models such as a contour model114, as illustrated in FIG. 1C, and a surface model 116 are used tocapture the real-world automotive radar measurements 112. However, theaforesaid spatial models are inaccurate. Some embodiments are based onthe recognition that real-world automotive radar measurements 112 aredistributed around edges of the objects (the vehicle 106) with a certainvolume, which gives rise to a surface volume model. To that end, someembodiments are based on objective of formulating a surface volume model118 that resembles and captures the real-world automotive radarmeasurements 112. The surface volume model 118 balances between thecontour model 114 and the surface model 116 with more realistic featureswhile keeping the EOT accurate.

In particular, in an embodiment, based on principles of the contourmodel 114 and the surface model 116, a compound measurement model 120(which is a type of surface volume model) is determined. The compoundmeasurement model 120 includes multiple probabilistic distributions 122that are geometrically constrained to a contour 124 of the object. InFIG. 1C, this geometrical constraint is that the centers of multipleprobabilistic distributions lies on the contour. The compoundmeasurement model then has a predetermined relative geometrical mappingto the center of the object. The multiple probabilistic distributions122 are used to cover a measurement spread along the contour 124 of theobject.

The compound measurement model 120 is compound in multiple ways. Forexample, the compound measurement model 120 has a compound structure,i.e., the multiple probabilistic distributions 122. Also, the compoundmeasurement model 120 has a compound composition, i.e., functions of themultiple probabilistic distributions 122, a function of the contour 124,and their relationship. Further, the compound measurement model 120 hasa compound nature, i.e., the multiple probabilistic distributions 122are based on measurements and thus represent data-driven approaches ofmodel generation, whereas the contour 124 is based on modeling a shapeof the object, e.g., a shape of a vehicle, using principles ofphysics-based modeling.

Additionally, the compound measurement model 120 takes advantage ofdifferent principles of modeling the expanded state, i.e., the compoundmeasurement model 120 joins the principles of the contour model 114 andthe surface model 116. As a result, the compound measurement model 120better represents a physical nature of tracking of the object whilesimplifying measurement assignment. In addition, the multipleprobabilistic distributions 122 of the compound measurement model 120are more flexible over a single distribution of the surface model 116,and may be configured to better describe the contour 124, and arefurther more flexible to explain the measurements coming from differentangles or views of the object.

Some embodiments are based on understanding that, in theory, themultiple probabilistic distributions 122 can lie on the contour 124,assuming that a shape of the contour 124 has no restrictions. However,in practice, such assumptions are incorrect and useless for tracking theexpanded state. In contrast, the contour 124 of the object ispredetermined and the multiple probabilistic distributions 122 are fitto the contour 124 rather than the contour 124 is fit to the multipleprobabilistic distributions 122. This allows reflecting a physicalstructure of the object during an update stage of the probabilisticfilter.

The compound measurement model 120 is learned offline, i.e., in advance.The compound measurement model 120 may be learned in a unit coordinatesystem or a global coordinate system. Some embodiments are based onrecognition that it is beneficial to learn the compound measurementmodel 120 in the unit coordinate system, because it simplifiescalculation and makes the compound measurement model 120 agnostic to thedimensions of the object. Each of the multiple probabilisticdistributions 122 (represented as ellipses) can be assigned withmeasurements in a probabilistic manner. The measurements associated withthe ellipse may be referred to as ellipse-assigned measurements.

Some embodiments are based on a recognition that the expanded state ofthe object can be tracked online, i.e., in real-time, using the compoundmeasurement model 120. Specifically, various embodiments track theexpanded state of the object using the probabilistic filter that tracksa belief on the expanded state of the object, wherein the belief on theexpanded state of the object is predicted using a motion model of theobject and is updated using the compound measurement model 120 of theobject.

FIG. 2 shows a block diagram of a tracking system 200 for tracking theexpanded state of the object by using the compound measurement model 120(shown in previous figures), according to some embodiments. The objectmay be a vehicle, such as, but not limited to, a car, bike, bus, ortruck. Also, the vehicle may be an autonomous or a semi-autonomousvehicle. The expanded state includes the kinematic state and theextended state of the object. According to some embodiments, thekinematic state corresponds to motion parameters of the object, such asvelocity, acceleration, heading and turn-rate. In some otherembodiments, the kinematic state corresponds to the position of theobject with its motion parameters. The tracking system 200 may include asensor 202 or be operatively connected to a set of sensors to probe ascene with one or multiple signal transmissions. The one or multiplesignal transmissions in turn are configured to produce one or multiplemeasurements of the object per transmission. According to someembodiments, the sensor 202 may be the automotive radar. In someembodiments, the scene includes a moving object. In some otherembodiments, the scene may include one or more objects that include bothmoving objects and stationary objects.

The tracking system 200 can have a number of interfaces connecting thetracking system 200 with other systems and devices. For example, anetwork interface controller (NIC) 214 is adapted to connect thetracking system 200 through a bus 212 to a network 216 connecting thetracking system 200 with a set of sensors. Through the network 216,either wirelessly or through wires, the tracking system 200 receivesdata of reflections of the one or multiple signal transmissions toproduce the one or multiple measurements of the object per transmission.Additionally or alternatively, the tracking system 200 includes anoutput interface 220 configured to submit control inputs to a controller222.

The tracking system 200 also includes a processor 204 configured toexecute stored instructions, as well as a memory 206 that storesinstructions that are executable by the processor 204. The processor 204can be a single core processor, a multi-core processor, a computingcluster, or any number of other configurations. The memory 206 caninclude random access memory (RAM), read only memory (ROM), flashmemory, or any other suitable memory systems. The processor 204 isconnected through the bus 212 to one or more input and output devices.Further the tracking system 200 includes a storage device 208 adapted tostore different modules including instructions executable by theprocessor 204. The storage device 208 can be implemented using a harddrive, an optical drive, a thumb drive, an array of drives, or anycombinations thereof.

The storage device 208 is configured to store a motion model 210 a ofthe object and a compound measurement model 210 b of the object (e.g.,the compound measurement model 120). The processor 204 is configured toexecute iteratively, a probabilistic filter, for iteratively tracking abelief on the expanded state of the object predicted using the motionmodel 210 a of the object and updated using the compound measurementmodel 210 b of the object. The tracking of the belief on the expandedstate of the object is described in detail below with reference to FIGS.3A, 3B, and 3C.

FIG. 3A shows a schematic for computing predicted measurements and acovariance matrix, according to some embodiments. Given an expandedstate 300 of the object and a covariance matrix corresponding to aprevious time step, and a motion model of the object, an expanded state302 of the object for a current time step and a covariance matrixcorresponding to the expanded state 302 are predicted by the processor204. The expanded state 300 of the object corresponding to the previoustime step is denoted as x_(k−1|k−1). The predicted expanded state 302 ofthe object is denoted as x_(k|k−1). The expanded state 300 includesvarious kinematic states, for example, x=[x_(m), y_(m), v, ψ, ω]^(T),where (x_(m), y_(m))^(T) is the center of the object, v is a polarvelocity of the vehicle, ψ is an orientation angle, and ω is a turningrate. In an alternate embodiment, the expanded state 300 includes theextended state in addition to the kinematic states, for example,x=[x_(m), y_(m), v, ψ, ω, l, w]^(T), where l and w are length and widthof the object, respectively. Likewise, the predicted expanded state 302includes predicted kinematic states and/or predicted extended states. Inan embodiment, the motion model may be a coordinated turn (CT) motionmodel with polar velocity. In some other embodiments, for the kinematicstates, the CT motion model with the polar velocity is used and, for theextended state, i.e., the length and width, a constant model is usedwith a process noise with a small covariance as the length and width areunlikely changed over time.

The predicted expanded state 302 of the object may be referred to as apredicted belief of the expanded state because this prediction isprobabilistic. Some embodiments are based on a recognition that thepredicted expanded state 302 of the object may be inaccurate to generatepredicted measurement for the expanded state as it requires an accuratespatial model of automotive radar measurements. To this end, in someembodiments, the compound measurement model 304 in a unit coordinatesystem that is learned offline is used. To align the compoundmeasurement model 304 in the unit coordinate system with the predictedexpanded state 302, the compound measurement model 304 needs to betransformed from the unit coordinate system to the global coordinatesystem with respect to the predicted expanded state 302. In particular,the ellipse-assigned measurements in the unit coordinate system need tobe transformed into the global coordinate system.

Some embodiments are based on realization that such a transformation canbe achieved using an unscented transform function 308. To that end, inan embodiment, the processor 204 generates sigma points for an ellipse306 (i.e., for a probabilistic distribution of the compound measurementmodel 304). The “ellipse” and “probabilistic distribution” may be usedinterchangeably and would mean the same. Further, the sigma points arepropagated into the unscented transform function 308 which is a functionof the predicted state 302 and, consequently, predicted measurements inthe global coordinate system corresponding to the ellipse-assignedmeasurements of the ellipse 306 in the unit coordinate system isdetermined. Additionally, a covariance corresponding to the predictedmeasurements is determined based on the predicted measurements.Likewise, the measurements in the global coordinate system correspondingto the ellipse-assigned measurements associated with the rest of theellipses are determined. To that end, a predicted expanded state model310, where the compound measurement model 304 is aligned according tothe predicted expanded state 302, is obtained. Further, syntheticmeasurements are determined for each probabilistic distribution of thepredicted expanded state model 310 as described below with reference toFIG. 3B.

FIG. 3B shows a schematic for determining the synthetic measurements foreach probabilistic distribution of the predicted expanded state model310, according to some embodiments. The processor 204 receivesmeasurements 312 (represented by cross marks) at the current time step.Some embodiments are based on the realization that the multipleprobabilistic distributions 314 a-314 h (ellipses) can be treatedindependently, e.g., parallelly to each other. Such an independenttreatment allows considering different view-angles of probing theexpanded state of the object. To consider such an independent treatment,some embodiments treat different probabilistic distributions of themultiple probabilistic distributions 314 a-314 h as belonging todifferent objects. Additionally, some embodiments are based on therealization that a soft probabilistic assignment, i.e., probabilisticassignment of the measurements 312 to different probabilisticdistributions, is more advantageous than a hard deterministicassignment. Soft probabilistic assignment may avoid catastrophicassignment of hard assignment while keeping the association dimensionlinear with respect to the number of ellipses and measurements.

To that end, processor 204 assigns the measurements 312 to theprobabilistic distribution 314 with an association probability.Likewise, processor 204 assigns the measurements 312 to each of theprobabilistic distributions 314 a-314 h with a corresponding associationprobability. The measurements with the corresponding associationprobability associated with each of multiple probabilistic distributions314 a-314 h is referred to as the ‘synthetic measurements’.

Further, for the probabilistic distribution 314 a, the processor 204determines, based on the synthetic measurements associated with theprobabilistic distribution 314 a, a synthetic centroid 316 a and asynthetic covariance matrix defining a spread 316 b. Likewise, for theprobabilistic distribution 314 e, the processor 204 determines, based onthe synthetic measurements associated with the probabilisticdistribution 314 e, a synthetic centroid 318 a and a syntheticcovariance matrix defining a spread 318 b. Likewise, for theprobabilistic distribution 314 h, the processor 204 determines, based onthe synthetic measurements associated with the probabilisticdistribution 314 h, a synthetic centroid 320 a and a syntheticcovariance matrix defining a spread 320 b. In such a way, the syntheticcentroid and the synthetic covariance matrix are determined for eachprobabilistic distribution. Further, using the synthetic measurementsassociated with each probabilistic distribution, the predicted belief onthe expanded state is updated as described below with reference to FIG.3C.

FIG. 3C shows a schematic for updating the predicted belief on theexpanded state 302, according to some embodiments. The processor 204updates the predicted belief on the expanded state 302 using theprobabilistic filter, such as a Kalman filter, with the syntheticmeasurements associated with each probabilistic distribution to producean updated expanded state x_(k|k) 322 of the object. The updatedexpanded state x_(k|k) 322 of the object may be referred to as anupdated belief on the expanded state. Further, the updated belief on theexpanded state is used to update the tracked belief. In an embodiment,the tracked belief is updated based on a difference between thepredicted belief and the updated belief. Further, the processor 204tracks the expanded state of the object based on the updated trackedbelief on the expanded state.

The compound measurement model 304 used for tracking the expanded stateof the object, as described above, is learned offline. The offlinelearning and characteristics of the compound measurement model 304 aredescribed below.

The compound measurement model 304 includes, for instance, L Gaussiancomponents (i.e., ellipses) with their component means located on thecontour. In an embodiment, the contour may be a B-spline curve. TheB-spline curve is advantageous because the B-spline curve provides morecontrol flexibility for enclosed contours. Also, since the B-splinecurves satisfy a strong convex hull property, they have a finer shapecontrol. For each ellipse centered at μ_(l) with an extent Σ_(l), N_(k)measurements may be assigned with an association probability ρ_(i) ^(l).Given a measurement-to-ellipse assignment, a likelihood function isgiven as

ϕ(?|?, ?, ?, ?) × 𝒩(?, ?) × w(? − ?)where? = ? $\begin{matrix}{{\text{?} = \text{?}},} & {(1)}\end{matrix}$ $\begin{matrix}{{\text{?} = {\text{?}\left( {\text{?} - \text{?}} \right)\left( {\text{?} - \text{?}} \right)^{T}}},} & {(2)}\end{matrix}$ ?indicates text missing or illegible when filed

(1) and (2) correspond to a sample mean and spread of l-th ellipse.

denotes a Gaussian distribution and

is a Wishart distribution.

Some embodiments are based on recognition that the probabilisticdistributions of the compound measurement model 304 can be representedusing Gaussian distribution to better align with probabilistic filters.For example, in some embodiments, the probabilistic distributions aredefined as a random matrix model (RMM) in a probability space (Ω, P, F)where the sample space Ω is a set of matrices. The random matrices areadvantageous to represent multi-dimensional probabilistic distributionsand parameters of the probabilistic distributions represented as RMMscan be illustrated using oval shapes. According to an embodiment, withall L ellipses and given the measurement-to-ellipse assignment, L randommatrices model is defined as

$\begin{matrix}{{{p\left( {\left. Z \middle| \theta \right.,\rho} \right)} = {\text{?}{\phi\left( {\left. \text{?} \middle| \text{?} \right.,\text{?},\text{?},\rho} \right)}}},} & {(3)}\end{matrix}$ ?indicates text missing or illegible when filed

where mixture weights π_(l) are assumed to equal π_(l)=1/L.

Further, it is assumed that the ellipse centers are located on aB-spline curve defined by c(r)∈

^(2×1) of degree d

$\begin{matrix}{{{c(r)} = {\text{?}(r)}},{0 \leq r \leq {m - d + 1}},} & {(4)}\end{matrix}$ ?indicates text missing or illegible when filed

where p_(j) ∈

^(2×1) is a j-th control point, m+1 is a number of control points, andB_(j,d)(r) is a basis function with a parameter r. By enforcingμ_(l)=c(r_(l)) with r_(l) denoting a corresponding parameter of the l-thellipse center μ_(l), a B-spline chained ellipses model (i.e., thecompound measurement model 304) is defined as

$\begin{matrix}{{{p\left( {\left. Z \middle| \theta \right.,\rho} \right)} = {\sum\limits_{l = 1}^{L}{\text{?}\left( {\left. \text{?} \middle| \text{?} \right.,{\text{?}\left( \text{?} \right)},\text{?},\rho} \right)}}},} & {(5)}\end{matrix}$ ?indicates text missing or illegible when filed

where parameters of the B-spline chained ellipses model (i.e., thecompound measurement model 304) are a number of measurements for eachcomponent N, the control points of the B-spline curve {p_(j)}_(j=0) ^(m)and the covariance matrices of each component {Σ_(l)}_(l=1) ^(L).

FIG. 4 shows a flow chart of a method for learning the parameters of thecompound measurement model 304, according to some embodiments. At step400, the method includes accepting 400 training data including differentmeasurements of different motions of different objects. At step 402, themethod includes transforming 402 the training data into a commoncoordinate system.

Some embodiments are based on recognition that the parameters of thecompound measurement model 304 can be learned offline based on thetraining data and knowledge of the contour of the object to be trackedusing various statistical methods, such as an expectation-maximization(EM) method. To that end, at step 404, the method includes learning 404the parameters of the compound measurement model from the training data,using the statistical method, such as the EM method.

FIGS. 5A and 5B show schematics of transformation of the training datacollected from different motions of different objects into the commonunit coordinate system, according to some embodiments. The differentmeasurements collected from tracking different trajectories 500 and 502are converted into a respective object-centered (OC) coordinate system504 and 506. Then, the converted measurements are aggregated 508. Insome implementations, the measurements are collected for motion ofsimilar type of objects, e.g., from motions of similar class ofvehicles. For example, the embodiments, for each trajectory, convert themeasurements from each time step from global coordinate (GC) to theobject-centered (OC) coordinate, and aggregate OC measurements from alltrajectories for vehicles with a similar size (e.g., sedan).

Next, as shown in FIG. 5B, the embodiments convert the aggregated OC 508measurements to a unit coordinate (UC) system 510. In someimplementations, the conversion to UC system is performed by variousnormalization techniques that allow using the converted training datafor machine learning. Further, the measurements in the unit coordinatesystem 510 are used as training data for learning the parameters of thecompound measurement model 304.

FIG. 6 shows a block diagram of the EM method for learning theparameters of the compound measurement model 304, according to someembodiments. In some embodiments, the measurements in the unitcoordinate system 510 are used as the training data 600, denoted asZ={z_(j)}_(i=1) ^(N), for the EM method. In some other embodiments, theaggregated OC measurements 508 are received and converted into a unitcoordinate system, which originates at the object center m=[x_(m),y_(m)]∈

^(2×1) and is oriented such that x-axis points towards object's front,by applying the following coordinate transformation:

{tilde over (z)} _(i) =S ⁻¹ R _({dot over (v)}) ⁻¹(z _(i) −m),  (6)

where R_(ψ)∈

^(2×2) is a rotation matrix as a function of the orientation angle ψ,S=diag(1, w) is a scaling matrix.

The training data 600 in the unit coordinate system, and initialparameters 602 such as a control point and extent I° are input data tothe EM method. The EM method includes two main steps, namely, anexpectation step 604 and a maximization step 606.

The expectation step 604 is to update hidden random variables {ρ_(l), z_(l), Z _(l),}. At first, a posterior association probability of eachmeasurement is calculated as

$\begin{matrix}{{\text{?} = \frac{\text{?} \times {\mathcal{N}\left( \text{?} \right)}}{{\text{?} \times {\sum_{l = 1}^{L}{\mathcal{N}\left( \text{?} \right)}}} + \lambda}},} & {(7)}\end{matrix}$ ?indicates text missing or illegible when filed

where μ_(i) are 4Σ_(j) are the mean and the covariance matrix of eachcomponent. Scaling factor 4 is used to approximate a uniformdistribution and λ is a probability of uniformly distributed outliers.Then, the remaining hidden variables z _(l), Z _(l) can be updated using(1) and (2), respectively.

The maximization step 606 is to update the model parameters θ={p_(j),Σ_(l)} based on the Q-function of (5) as

$\begin{matrix}{{Q(\theta)}\text{?}\text{?}{\left\{ {{{- \text{?}}\left( {\text{?} - \text{?}} \right)^{T}\text{?}\left( {\text{?} - \text{?}} \right)} - {\text{?}\log{❘\sum_{l}❘}} - {\frac{1}{2}{{tr}\left( {{- \frac{1}{2}}\text{?}\text{?}} \right)}}} \right\}.}} & (8)\end{matrix}$ ?indicates text missing or illegible when filed

The B-spline curve in a matrix-vector form can be reformatted asμ_(i)=B_(l)p, where

B_(l)=blkding(n_(l) ^(T),n_(l) ^(T)),n_(l)=[B_(n,d)(n), . . .B_(m,d)(r_(i))]^(T) and p=, [p_(x) ^(T),p_(y) ^(T)] with p_(x) ^(T) andp_(y) ^(T) denoting the control inputs in x and y coordinates,respectively. By setting derivative of Q (θ) (with respect to θ) to 0,the control input can be given as p=H⁺M, where H⁺ is Moore-Penroseinverse of H=Σ_(l=1) ^({circumflex over (L)})(N ^(l)B_(l) ^(T)Σ_(l)⁻¹B_(l)) and M=Σ_(l=1) ^(L)(Ñ_(l)B_(l) ^(T)Σ_(l) ⁻¹ z _(l)), and

$\begin{matrix}{\sum_{l}{= {{\frac{1}{\text{?} + 1}\left\lbrack {{\text{?}\left( {\text{?} - \text{?}} \right)\left( {\text{?} - \text{?}} \right)^{T}} + {\overset{\_}{Z}}_{l}^{T}} \right\rbrack}.}}} & (9)\end{matrix}$ ?indicates text missing or illegible when filed

Further, iterations are carried out between the estimates of p and untila convergence criterion 608 is achieved. The convergence criterion 608may be a predetermined likelihood in (8), relative changes of theestimated parameters over consecutive iterations is smaller thanpredefined values, or a predetermined maximum number of iterations.

According to some embodiments, the offline learned compound measurementmodel is used for online tracking of the expanded state of the object,i.e., real time tracking of the expanded state of the object. Someembodiments are based on the realization that the probabilistic natureof the compound measurement model can be beneficially aligned withprobabilistic multi-hypothesis tracking (PMHT) methods. For example,such an alignment allows implementing the probabilistic filter using atleast a variation of a Kalman filter. For example, one embodiment usesan unscented Kalman filter-probabilistic multi-hypothesis tracking(UKF-PMHT) method. The unscented Kalman filter (UKF) is used fortransforming the compound measurement model from the unit coordinatesystem into the global coordinate system. The probabilisticmulti-hypothesis tracking (PMHT) method is then applied to assign themeasurements at the current time step to different ellipsis componentsin a probabilistic fashion, and update the expanded state of the object.

In an embodiment, given the offline learned compound measurement modeland assuming a measurement x_(μ) in the unit coordinate system isdistributed with respect to the 1-th ellipse

(μ_(l), Σ_(l)), the corresponding measurement h_(l,k)(x_(k|k−1)) in theglobal coordinate system is defined as

h _(l,k)(x _(k|k−1))=m _(k|k−1) +R _(φk|k−1)·8_(k|k−1) ·x _(μ)  (10)

where m_(ψk|k−1), R_(ψk|k−1) and s=diag(l_(k|k−1), w_(k|k−1)) aredefined the same way as (6) except that all augments are given by thepredicted state (k|k−1) with corresponding predictive distributions(e.g., the Gaussian distribution).

Some embodiments are based on realization that since the transformationin (10) is nonlinear, particularly with respect to the predictiveorientation angle, an unscented transform (UT) can be used to determinea mean h _(l,k)(x_(k|k−1)) and a covariance matrix X_(l) ofh_(l,k)(x_(k|k−1)). To this end, the predicted expanded state isaugmented with x_(μ) as x_(aug)=[x_(k|k−1), x_(μ)]^(T)

^(n) ^(a) ^(×1) with n_(a)=9. Then, 2n_(a)+1 weighted samples, i.e., thesigma points, are determined such that they can describe a true mean x_(aug) and a covariance matrix P_(aug) of x_(aug):

_(n) =x _(aug) ,

n=k/(n _(a) +k),

_(i≥1)=0.5(n _(a)+κ),

_(i≤n) _(a) =x _(aug)+(√{square root over ((n _(a)+κ)P_(aug))})_(i),  (11)

_(i≤n) _(a) =x _(aug)+(√{square root over ((n _(a)+κ)P _(aug))})_(i−n)_(a) ,  (12)

where κ is a scaling parameter such thāt κ+n_(a)≠0 and (√{square rootover (A)})_(i) denotes i-th row of matrix square root of A. Each sigmapoint is then propagated through the nonlinear function of (10), i.e.,

=h_(l,k)(

), and a first two moments of h_(l,k)(x_(k|k−1)) are computed as

$\begin{matrix}{{\text{?} = {\text{?}W_{i}B_{i}}},} & {(13)}\end{matrix}$ $\begin{matrix}{Χ_{l} = {\text{?}{W_{i}\left( {B_{i} - {\overset{\_}{h}}_{l,k}} \right)}{\left( {B_{i} - {\overset{\_}{h}}_{l,k}} \right)^{T}.}}} & {(14)}\end{matrix}$ ?indicates text missing or illegible when filed

In the global coordinate system, a measurement z_(i) that is assigned tothe l-th ellipse can be defined as

z _(i) =h _(l,k)(x _(k|k−1))+n _(l)  (15)

where h_(l,k)(x_(k|k−1))˜

(h _(l,k), X_(l)), is a corresponding reflection center, and n_(l)˜

(0,R) is a measurement noise.

Given measurements at time k, Z_(k)={z_(i,k)}_(i=1) ^(N) ^(k) and Loffline learned {h_(l,k)(x_(k|k−1))}_(i=1) ^(L) obtained at time step k,the PMHT is used to assign the measurements to each ellipse component.Different from a general PMHT algorithm to handle measurement-to-objectassociation and update the kinematic states of multiple objects overconsecutive time steps, the PMHT algorithm here is applied to handle themeasurement-to-ellipsis association and update both kinematic and extentstates of a single object over the current time step k. Specifically,the PMHT employs the EM algorithm for a soft measurement-to-ellipsisassignment that in turns creates a synthetic measurement for eachcomponent. Mathematically, the measurement-to-ellipsis associationweights ρ_(i,k) ^(l), synthetic measurements Z _(l,k) and correspondingsynthetic covariance matrix C_(zz) are derived as follows

$\begin{matrix}{{\rho_{i,k}^{l} = \frac{\mathcal{N}\left( {{\text{?}\left( \text{?} \right)},{\text{?} + R}} \right)}{\sum_{i = 1}^{L_{k}}{\mathcal{N}\left( {{\text{?}\left( \text{?} \right)},{\text{?} + R}} \right)}}},} & {(16)}\end{matrix}$ $\begin{matrix}{{{\overset{\_}{z}}_{l,k} = \frac{\sum_{i = 1}^{N_{k}}{\rho_{i,k}^{l}\text{?}}}{\sum_{i = 1}^{N_{k}}\rho_{i,k}^{l}}},} & {(17)}\end{matrix}$ $\begin{matrix}{C_{xz} = {{4Χ_{l,k}} + {\frac{R}{\text{?}\rho_{i,k}^{l}}.}}} & {(18)}\end{matrix}$ ?indicates text missing or illegible when filed

Further, a covariance between the expanded state and measurements C_(xz)is calculated during the UT procedure (10) and the filter gain iscalculated as K=C_(xz) C_(zz) ⁻¹. The expanded state x_(k,l) and thecovariance matrix C_(l,n) are updated based on the l-th measurementequation in (15). The PMHT iterates between the expectation andmaximization steps until a predefined maximum iteration number N_(iter)is reached. In each iteration n, the expanded state x_(k,l) and thecovariance matrix C_(l,n) are updated incrementally by each component(i.e., over l) in order of (10) and (16)-(18). An overall UKF-PMHTtracking algorithm is described below.

FIG. 7A shows a flowchart of the UKF-PMHT tracking algorithm, accordingto some embodiments. The UKF-PMHT tracking algorithm is executed by theprocessor 204. The UKF-PMHT tracking algorithm includes two stages,namely, a prediction stage 700 and an update stage 704. In theprediction stage 700, a belief on the expanded state of the object and acorresponding covariance matrix is predicted using the motion model.Further, at block 702, an iteration of the update stage 704 is initiatedby setting an iteration index n=1, and x_(l,1)=x_(k|k−1) andC_(l,1)=C_(k|k−1).

At block 706 of the update stage 704, the predicted measurements and thecovariance matrix are computed based on an offline learned measurementmodel 712. FIG. 7B shows a block diagram of steps performed to computethe predicted measurements and the covariance matrix, according to someembodiments. At block 720, the sigma points

_(i=1) ^(n) ^(a) are generated for a first ellipse (i.e., l=1) of thecompound measurement model 712 and the latest updated expanded state,using equations (11) and (12). At block 722, the sigma points arepropagated through an affine transform given by (10). At block 724, thepredicted measurements and the covariance matrix are computed accordingto equations (13) and (14), respectively, for the first ellipse.

Referring back to FIG. 7A, at block 708, given measurements 714 at timek, the synthetic measurements and the synthetic covariance matrix arecomputed. FIG. 7C shows a block diagram of steps performed to computethe synthetic measurements and the synthetic covariance matrix,according to some embodiments. At block 726, the measurements-to-ellipseassociation weight is computed according to equation (16) for the firstellipse. At block 728, the synthetic measurements and the syntheticcovariance matrix are computed using equations (17) and (18),respectively, for the first ellipse.

Referring back to FIG. 7A, at block 710, the expanded state x_(l,n) andthe covariance matrix C_(l,n) are updated. Here, subscript l is anellipse index. FIG. 7D shows a block diagram of steps performed toupdate the expanded state and the covariance matrix C_(l,n) for thefirst ellipse (i.e., l=1), according to some embodiments. At block 730,the cross-covariance matrix is computed as

$C_{{xz},l} = {\sum\limits_{i = 0}^{2n_{a}}{{{W_{i}\left\lbrack {{A_{i}\left( x_{k|{k - 1}} \right)} - x_{k|{k - 1}}} \right\rbrack}\left\lbrack {{B_{i}\left( x_{\mu,l} \right)} - {\overset{\_}{h}}_{l,k}} \right\rbrack}^{T}.}}$

At block 732, a Kalman filter gain is computed. The Kalman filter gainis given by

K _(l) =C _(xz,l) C _(zz,l) ⁻¹

At block 734, the expanded state x_(l,n) and the covariance matrix areupdated as

x _(l,n) =x _(l−1,n) +K _(l)( z _(l,k) −h _(l,k))

C _(l,n) =C _(l−1,n) −K _(l) C _(zz,l) K _(l) ^(T)

Further, the same functions given at the blocks 706, 708, and 710 areexecuted for a second ellipse (i.e., l=2) of the compound measurementmodel 712. For the second ellipse, the updated expanded state x_(l,n)and the covariance matrix C_(l,n) are used to compute the predictedmeasurements and the covariance matrix. In other words, the predictedmeasurements and the covariance matrix are computed using the latestupdated expanded state and the covariance matrix. Likewise, the samefunctions given at the blocks 706, 708, and 710 are executed for therest of ellipses of the compound measurement model 712. To that end, tocomplete an iteration of the update stage 704, a number of internaliterations l=1 . . . L, where L is a number of ellipses of the compoundmeasurement model 712, are executed. The complete execution of theiteration (n=1) of the update stage 704 yields an expanded state of theobject at time k. Further, in the next iteration (i.e., n=2), the updatestage 704 is executed to yield an updated expanded state. The updatestage 704 is iteratively executed until a convergence criterion 716 isachieved. The convergence criterion 710 may be a predetermined maximumnumber of iterations N_(iter). Once the convergence criterion 716 isachieved, an updated expanded state

x_(k|k) = x_(L, N_(iter))

and a corresponding covariance matrix

C_(k|k) = C_(L, N_(iter))

are outputted 718.

FIG. 8A shows a schematic of a vehicle 800 including a controller 802 incommunication with the tracking system 200 employing principles of someembodiments. The vehicle 800 may be any type of wheeled vehicle, such asa passenger car, bus, or rover. Also, the vehicle 800 can be anautonomous or semi-autonomous vehicle. For example, some embodimentscontrol the motion of the vehicle 800. Examples of the motion includelateral motion of the vehicle controlled by a steering system 804 of thevehicle 800. In one embodiment, the steering system 804 is controlled bythe controller 802. Additionally or alternatively, the steering system804 may be controlled by a driver of the vehicle 800.

In some embodiments, the vehicle may include an engine 810, which can becontrolled by the controller 802 or by other components of the vehicle800. In some embodiments, the vehicle may include an electric motor inplace of the engine 810 and can be controlled by the controller 802 orby other components of the vehicle 800. The vehicle can also include oneor more sensors 806 to sense the surrounding environment. Examples ofthe sensors 806 include distance range finders, such as radars. In someembodiments, the vehicle 800 includes one or more sensors 808 to senseits current motion parameters and internal status. Examples of the oneor more sensors 808 include global positioning system (GPS),accelerometers, inertial measurement units, gyroscopes, shaft rotationalsensors, torque sensors, deflection sensors, pressure sensor, and flowsensors. The sensors provide information to the controller 802. Thevehicle may be equipped with a transceiver 812 enabling communicationcapabilities of the controller 802 through wired or wirelesscommunication channels with the tracking system 200 of some embodiments.For example, through the transceiver 812, the controller 802 receivesthe control inputs from the tracking system 200.

FIG. 8B shows a schematic of interaction between the controller 802 andcontrollers 814 of the vehicle 800, according to some embodiments. Forexample, in some embodiments, the controllers 814 of the vehicle 800 aresteering control 816 and brake/throttle controllers 818 that controlrotation and acceleration of the vehicle 800. In such a case, thecontroller 802 outputs control commands, based on the control inputs, tothe controllers 816 and 818 to control the kinematic state of thevehicle. In some embodiments, the controllers 814 also includeshigh-level controllers, e.g. a lane-keeping assist controller 820 thatfurther process the control commands of the controller 802. In bothcases, the controllers 814 utilize the output of the controller 802 i.e.control commands to control at least one actuator of the vehicle, suchas the steering wheel and/or the brakes of the vehicle, in order tocontrol the motion of the vehicle.

FIG. 8C shows a schematic of an autonomous or semi-autonomous controlledvehicle 822 for which the control inputs are generated by using someembodiments. The controlled vehicle 822 may be equipped with thetracking system 200. In some embodiments, an expanded state of each ofthe obstacles 824 are tracked by the controlled vehicle 822 andsubsequently, the control inputs are generated based on the trackedexpanded states of the obstacles. In some embodiments, the controlinputs include commands specifying values of one or combination of asteering angle of the wheels of the vehicle and a rotational velocity ofthe wheels, and the measurements include values of one or combination ofa rotation rate of the vehicle and an acceleration of the vehicle.

The generated control inputs aim to keep the controlled vehicle 822within particular bounds of road 826, and aims to avoid otheruncontrolled vehicles, i.e., obstacles 824 for the controlled vehicle822. For example, based on the control inputs, the autonomous orsemi-autonomous controlled vehicle 822 may, for example, pass anothervehicle on the left or on the right side or instead to stay behindanother vehicle within the current lane of the road 826.

The following description provides exemplary embodiments only, and isnot intended to limit the scope, applicability, or configuration of thedisclosure. Rather, the following description of the exemplaryembodiments will provide those skilled in the art with an enablingdescription for implementing one or more exemplary embodiments.Contemplated are various changes that may be made in the function andarrangement of elements without departing from the spirit and scope ofthe subject matter disclosed as set forth in the appended claims.

Specific details are given in the following description to provide athorough understanding of the embodiments. However, understood by one ofordinary skill in the art can be that the embodiments may be practicedwithout these specific details. For example, systems, processes, andother elements in the subject matter disclosed may be shown ascomponents in block diagram form in order not to obscure the embodimentsin unnecessary detail. In other instances, well-known processes,structures, and techniques may be shown without unnecessary detail inorder to avoid obscuring the embodiments. Further, like referencenumbers and designations in the various drawings indicate like elements.

Also, individual embodiments may be described as a process which isdepicted as a flowchart, a flow diagram, a data flow diagram, astructure diagram, or a block diagram. Although a flowchart may describethe operations as a sequential process, many of the operations can beperformed in parallel or concurrently. In addition, the order of theoperations may be re-arranged. A process may be terminated when itsoperations are completed, but may have additional steps not discussed orincluded in a figure. Furthermore, not all operations in anyparticularly described process may occur in all embodiments. A processmay correspond to a method, a function, a procedure, a subroutine, asubprogram, etc. When a process corresponds to a function, thefunction's termination can correspond to a return of the function to thecalling function or the main function.

Furthermore, embodiments of the subject matter disclosed may beimplemented, at least in part, either manually or automatically. Manualor automatic implementations may be executed, or at least assisted,through the use of machines, hardware, software, firmware, middleware,microcode, hardware description languages, or any combination thereof.When implemented in software, firmware, middleware or microcode, theprogram code or code segments to perform the necessary tasks may bestored in a machine readable medium. A processor(s) may perform thenecessary tasks.

Various methods or processes outlined herein may be coded as softwarethat is executable on one or more processors that employ any one of avariety of operating systems or platforms. Additionally, such softwaremay be written using any of a number of suitable programming languagesand/or programming or scripting tools, and also may be compiled asexecutable machine language code or intermediate code that is executedon a framework or virtual machine. Typically, the functionality of theprogram modules may be combined or distributed as desired in variousembodiments.

Embodiments of the present disclosure may be embodied as a method, ofwhich an example has been provided. The acts performed as part of themethod may be ordered in any suitable way. Accordingly, embodiments maybe constructed in which acts are performed in an order different thanillustrated, which may include performing some acts concurrently, eventhough shown as sequential acts in illustrative embodiments.

Although the present disclosure has been described with reference tocertain preferred embodiments, it is to be understood that various otheradaptations and modifications can be made within the spirit and scope ofthe present disclosure. Therefore, it is the aspect of the appendedclaims to cover all such variations and modifications as come within thetrue spirit and scope of the present disclosure.

We claim:
 1. A tracking system for tracking an expanded state of an object, the expanded state including a kinematic state indicative of one or a combination of a position and a velocity of a center of the object, and an extended state indicative of one or a combination of a dimension and an orientation of the object, the tracking system comprising: at least one processor; and a memory having instructions stored thereon that, when executed by the at least one processor, cause the tracking system to: receive measurements associated with at least one sensor, wherein the at least one sensor is configured to probe a scene including the object via one or multiple signal transmissions, the one or multiple signal transmissions configured to produce one or multiple measurements of the object per the transmission; execute a probabilistic filter iteratively tracking a belief on the expanded state of the object, wherein the belief is predicted using a motion model of the object and is updated using a compound measurement model of the object, wherein the compound measurement model includes multiple probabilistic distributions constrained to lie on a contour of the object with a predetermined relative geometrical mapping to the center of the object, wherein in each iteration of the iterative tracking, the belief on the expanded state is updated based on a difference between a predicted belief and an updated belief, wherein the updated belief is estimated based on probabilities of the measurements fitting each of the multiple probabilistic distributions, and mapped to the expanded state of the object based on the corresponding geometrical mapping; and track the expanded state of the object based on the updated belief on the expanded state.
 2. The tracking system of claim 1, wherein the processor is further configured to transform the compound measurement model from a unit coordinate system to a global coordinate system with respect to the predicted belief or iteratively updated belief on the expanded state, based on an unscented transform function.
 3. The tracking system of claim 1, wherein the processor is further configured to assign the multiple measurements to different probabilistic distributions of the multiple probabilistic distributions by treating the different probabilistic distributions as belonging to different objects.
 4. The tracking system of claim 3, wherein the processor is further configured to assign the multiple measurements to the different probabilistic distributions using probabilistic multiple-hypothesis tracking (PMHT) for performing the assigning of the multiple measurements to the different probabilistic distributions treated as the different objects.
 5. The tracking system of claim 1, wherein the compound measurement model is learned based on an expectation-maximization (EM) method.
 6. The tracking system of claim 1, wherein one or more parameters of each probabilistic distribution are represented by a random matrix model (RMM) in a two-dimensional (2D) probability space.
 7. The tracking system of claim 1, wherein the contour of the object corresponds to a B-spline curve.
 8. The tracking system of claim 1, wherein the predicted belief on the state of the vehicle is used to align the compound measurements model with the measurements using an unscented transformation.
 9. The tracking system of claim 1, wherein the processor is configured to determine a control input to a controller of a vehicle based on the tracked expanded state of the object, and control the vehicle according to the control input; and wherein the vehicle is operatively connected to the tracking system of claim
 1. 10. A tracking method for tracking an expanded state of an object, wherein the expanded state includes a kinematic state indicative of one or a combination of a position and a velocity of a center of the object and an extended state indicative of one or a combination of a dimension and an orientation of the object, the method comprising: receiving measurements associated with at least one sensor, wherein the at least one sensor is configured to probe a scene including the object with one or multiple signal transmissions to produce one or multiple measurements of the object per the transmission; executing a probabilistic filter iteratively tracking a belief on the expanded state of the object, wherein the belief is predicted using a motion model of the object and is updated using a compound measurement model of the object, wherein the compound measurement model includes multiple probabilistic distributions constrained to lie on a contour of the object with a predetermined relative geometrical mapping to the center of the object, wherein in each iteration of the iterative tracking, the belief on the expanded state is updated based on a difference between a predicted belief and an updated belief, wherein the updated belief is estimated based on probabilities of the measurements fitting each of the multiple probabilistic distributions, and mapped to the expanded state of the object based on the corresponding geometrical mapping; and tracking the expanded state of the object based on the updated belief on the expanded state.
 11. The tracking method of claim 10, wherein the tracking method further comprises transforming the compound measurement model from a unit coordinate system to a global coordinate system with respect to the predicted belief on expanded state, based on an unscented transform function.
 12. The tracking method of claim 10, wherein the tracking method further comprises assigning the multiple measurements to different probabilistic distributions of the multiple probabilistic distributions independently from each other by treating the different probabilistic distributions as belonging to different objects.
 13. The tracking method of claim 12, wherein the tracking method further comprises assigning the multiple measurements to the different probabilistic distributions using probabilistic multiple-hypothesis tracking (PMHT) performing the assigning of the multiple measurements to the different probabilistic distributions treated as different objects.
 14. The tracking method of claim 10, wherein the compound measurement model is learned based on an expectation-maximization (EM) method.
 15. The tracking method of claim 10, wherein one or more parameters of each probabilistic distribution are represented by a random matrix model (RMM) in a two-dimensional (2D) probability space.
 16. The tracking method of claim 10, wherein the contour of the object corresponds to a B-spline curve.
 17. A non-transitory computer-readable storage medium embodied thereon a program executable by a processor for performing a method for tracking an expanded state of an object, wherein the expanded state includes a kinematic state indicative of one or a combination of a position and a velocity of a center of the object and an extended state indicative of one or a combination of a dimension and an orientation of the object, the method comprising: receiving measurements associated with at least one sensor, wherein the at least one sensor is configured to probe a scene including the object with one or multiple signal transmissions to produce one or multiple measurements of the object per the transmission; executing a probabilistic filter iteratively tracking a belief on the expanded state of the object, wherein the belief is predicted using a motion model of the object and is updated using a compound measurement model of the object, wherein the compound measurement model includes multiple probabilistic distributions constrained to lie on a contour of the object with a predetermined relative geometrical mapping to the center of the object, wherein in each iteration of the iterative tracking, the belief on the expanded state is updated based on a difference between a predicted belief and an updated belief, wherein the updated belief is estimated based on probabilities of the measurements fitting each of the multiple probabilistic distributions and mapped to the expanded state of the object based on the corresponding geometrical mapping; and tracking the expanded state of the object based on the updated belief on the expanded state. 